Orbits and Manifolds near the Equilibrium Points around a Rotating
Asteroid
Yu Jiang1, 2, Hexi Baoyin2, Junfeng Li2
1. State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China
2. School of Aerospace, Tsinghua University, Beijing 100084, China
Y. Jiang () e-mail: [email protected] (corresponding author)
H. Baoyin () e-mail: [email protected] (corresponding author)
Abstract. We study the orbits and manifolds near the equilibrium points of a rotating
asteroid. The linearised equations of motion relative to the equilibrium points in the
gravitational field of a rotating asteroid, the characteristic equation and the stable
conditions of the equilibrium points are derived and discussed. First, a new metric is
presented to link the orbit and the geodesic of the smooth manifold. Then, using the
eigenvalues of the characteristic equation, the equilibrium points are classified into 8
cases. A theorem is presented and proved to describe the structure of the submanifold
as well as the stable and unstable behaviours of a massless particle near the
equilibrium points. The linearly stable, the non-resonant unstable, and the resonant
equilibrium points are discussed. There are three families of periodic orbits and four
families of quasi-periodic orbits near the linearly stable equilibrium point. For the
non-resonant unstable equilibrium points, there are four cases; for the periodic orbit
and the quasi-periodic orbit, the structures of the submanifold and the subspace near
the equilibrium points are studied for each case. For the resonant equilibrium points,
the dimension of the resonant manifold is greater than four, and we find at least 1
family of periodic orbits near the resonant equilibrium points. Besides, this theory is
1
applied to asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia, and 6489
Golevka.
Key words: Asteroids; Equilibrium points; Stability; Periodic orbits; Smooth
manifold;
1. Introduction
Recently, several sample return missions to Near-Earth Asteroid (NEA) are selected
(Barucci 2012; Barucci et al. 2012a; Barucci et al. 2012b; Boynton et al. 2012;
Brucato 2012; Duffard et al. 2011), including MarcoPolo-R, OSIRIS-Rex, etc.
MarcoPolo-R is a sample return mission to a primitive Near-Earth Asteroid (NEA)
selected for the assessment study in the framework of ESA Cosmic Vision 2015-25
program (Barucci 2012; Barucci et al. 2012a; Barucci et al. 2012b; Brucato 2012).
The baseline target of MarcoPolo-R is a binary asteroid (175706) 1996 FG3, which
offers a very efficient operational and technical mission profile (Brucato 2012).
OSIRIS-Rex is an asteroid sample-return mission that was selected by NASA in May
2011 as the third New Frontiers Mission (Boynton et al. 2012).
MarcoPolo-R can help one to understand the unique geomorphology, dynamics
and evolution of a binary NEA (Barucci et al. 2012a). These missions make the
dynamics of a spacecraft (which is modeled as a massless particle hereafter) near
asteroids becomes an interesting topic. By applying the classic method of Legendre
polynomial series to the gravity potential of asteroids, the gravity potential may
diverge at some points in the gravitational field due to the irregular shape of the
2
asteroid (Balmino 1994; Elipe & Lara 2003). Several methods have been considered
to solve this difficulty. Some simple-shaped bodies have been used to simulate the
motion of a particle moving near an asteroid as well as to help understanding the
equilibrium points and the periodic orbits near asteroids. The dynamics around a
straight segment (Riaguas et al. 1999; Elipe & Lara 2003), a solid circular ring
(Broucke & Elipe 2005), a homogeneous annulus disk (Angelo & Claudio 2007;
Fukushima 2010), and a homogeneous cube (Liu et al. 2011) have been studied in
detail. These studies are limited to simple and symmetric bodies, which rotate around
their symmetric axes. For these bodies, the characteristic equation at the equilibrium
points is a quartic equation (Liu et al. 2011). Werner & Scheeres (1996) modelled the
shape and the gravity potential of the asteroids using homogeneous polyhedrons and
applied this method to 4769 Castalia. The polyhedron model is more precise than the
simple-shaped-body models and the Legendre polynomial model. However, this
model presents many free parameters (Elipe & Lara 2003; Werner & Scheeres 1996).
The dynamic equation of a massless particle near a rotating asteroid in the
body-fixed frame was studied and applied to 4769 Castalia and 4179 Toutatis by
Scheeres et al. (1996, 1998). The zero-velocity surface is defined using the Jacobi
integral, and this surface separates the forbidden regions from the allowable regions
for the particle (Scheeres et al. 1996; Yu & Baoyin 2012a). The periodic orbit families
near the asteroids can be calculated using Poincaré maps and numerical iterations
(Scheeres et al. 1996, 1998; Yu & Baoyin 2012b). The positions of the equilibrium
points near the asteroids as well as the eigenvalues and the periodic orbits near the
3
equilibrium points can be numerically calculated (Yu & Baoyin 2012a).
The stability of the equilibrium points was discussed by several authors using
special models. The periodic orbits of a particle around a massive straight segment
were investigated in Riaguas et al. (1999). A finite straight segment with constant
linear mass density was considered to study the chaotic motion and the equilibrium
points around the asteroid (433) Eros (Elipe & Lara 2003). The dynamics of a particle
around a homogeneous annulus disk was analysed in Alberti & Vidal (2007).
Moreover, a precise numerical method to calculate the acceleration vector that is
caused by a uniform ring or disk was described in Fukushima et al. (2010). The
dynamics of a particle around a non-rotating 2nd-degree and order-gravity field was
examined, and the closed-form solutions for the averaged orbital motion of the
particle were derived (Scheeres & Hu 2001).
This paper aims to derive the characteristic equation of the equilibrium points
near a rotating asteroid and to provide some corollaries about the stability, the
instability, and the resonance of the equilibrium points. Then, we will discuss the
orbits and the structure of the submanifold and the subspace near the equilibrium
points. This work can reveal the dynamics near the equilibrium points of a rotating
asteroid while providing the theoretical basis for orbital design and control of
spacecraft motion near an asteroid.
The linearised equations of motion that are relative to the equilibrium points in
the gravitational field of a rotating asteroid are derived in section 2; then, the
characteristic equation of the equilibrium points is presented. The stability conditions
4
of the equilibrium points in the potential field of a rotating asteroid are obtained and
proved in section 3. Furthermore, a new metric in a smooth manifold is provided in
Section 4 to link the orbit and the geodesic of the smooth manifold. The equilibrium
points are classified into eight cases using the eigenvalues of the characteristic
equation in section 5. It is found that the structure of the submanifold near the
equilibrium point is related to the eigenvalues. A theorem that describes the structure
of the submanifold and the stable and unstable behaviours of the particle near the
equilibrium points is proved. Using this theorem, the linearly stable, the non-resonant
unstable and the resonant equilibrium points are studied. The equilibrium point is
resonant if and only if at least two pure imaginary eigenvalues are equal.
Subsequently, four families of quasi-periodic orbits are found near the linearly stable
equilibrium point. The periodic orbit, the quasi-periodic orbit, the structure of the
submanifold and the subspace near the non-resonant unstable equilibrium points are
studied. For the resonant equilibrium points, there are three cases. For each case, the
resonant orbits and the resonant manifolds are analysed. There is at least one family
of periodic orbits near each resonant equilibrium point, and the dimension of the
resonant manifold is greater than four.
The theory is applied to asteroids 216 Kleopatra, 1620 Geographos, 4769
Castalia, and 6489 Golevka. For the asteroid 216 Kleopatra, 1620 Geographos, and
4769 Castalia, the equilibrium points are unstable, which are denoted as E1, E2, E3,
and E4. There are two families of periodic orbits and one family of quasi-periodic
orbits on the central manifold near each of the equilibrium points E1 & E2, while
5
there is only one family of periodic orbits on the central manifold near each of the
other two equilibrium points E3 & E4.
For the asteroid 6489 Golevka, two of the equilibrium points are unstable, which
are denoted as E1 and E2; while the other two of the equilibrium points are linearly
stable, which are denoted as E3 and E4. There are two families of periodic orbits as
well as one family of quasi-periodic orbits on the central manifold near each of the
two unstable equilibrium points E1 and E2, while there are three families of periodic
orbits as well as four families of quasi-periodic orbits on the central manifold near
each of the two linearly stable equilibrium points E3 and E4.
2. Equations of Motion
The equation of motion relative to the rotating asteroid can be given as (Scheeres
et al. 1996)
r  2ω×r  ω× ω×r   ω  r 
where
U  r 
 0,
r
(1)
is the body-fixed vector from the centre of mass of the asteroid to the
particle, ω is the rotational angular velocity vector of the asteroid relative to the
inertial frame of reference, and U  r  is the gravitational potential.
Let us define a function H as (Scheeres et al. 1996)
1
1
H  r  r   ω  r  ω  r   U  r 
2
2
(2)
If ω is time-invariant, then H is also time-invariant, and it is called the Jacobian
constant.
The effective potential can be defined as (Scheeres et al. 1996; Yu & Baoyin
6
2012a)
V r   
1
 ω  r  ω  r   U  r 
2
(3)
Hence, the equation of motion can be written as
r  2ω×r  ω  r 
V  r 
0
r
(4)
For a uniformly rotating asteroid, equation (4) can be written as (Yu & Baoyin 2012a)
r  2ω×r 
V  r 
 0,
r
(5)
and the Jacobian constant can be given by
1
H  r  r  V r 
2
(6)
The zero-velocity manifolds are determined by the following equation (Scheeres et al.
1996; Yu & Baoyin 2012a)
V r   H
(7)
The inequality V  r   H denotes the forbidden region for the particle, whereas
the inequality V  r   H denotes the allowable region for the particle. The equation
V  r   H implies that the velocity of the particle relative to the rotating body-fixed
frame is zero.
Let ω be the modulus of the vector ω; then, the unit vector e z is defined by
ω  ez . The body-fixed frame is defined through a set of orthonormal right-hand
unit vectors e
e x 
 
e  e y 
 
e z 
(8)
7
The equilibrium points are the critical points of the effective potential V  r  .
Thus, the equilibrium points satisfy the following condition
V  x, y, z  V  x, y, z  V  x, y, z 


 0,
x
y
z
where
 x, y, z 
 xL , yL , zL 
T
(9)
are the components of r in the body-fixed coordinate system. Let
denote the coordinates of a critical point; the effective potential
V  x, y, z  can be expanded using Taylor expansion at the equilibrium point
 xL , yL , zL 
T
. To study the stability of the equilibrium point, the equations of motion
relative to the equilibrium point are linearised, and the characteristic equation of
motion is derived. In addition, it is necessary to check whether any solutions of the
characteristic equation have positive real components. The Taylor expansion of the
effective potential V  x, y, z  at the point
 xL , yL , zL 
T
can be written as
1   2V 
1   2V 
1   2V 
2
2
2
V  x, y, z   V  xL , yL , zL    2   x  xL    2   y  yL    2   z  zL 
2  x L
2  y L
2  z L
  2V 
  2V 
  2V 

x

x
y

y

x

x
z

z

 
 
  y  yL  z  zL  
L 
L 
L 
L 
 xy L
 xz L
 yz L
(10)
where the solving derivation sequence is changeable, which implies that
 2V
 2V
 2V
 2V
 2V
 2V



,
,
.
xy yx xz zx yz zy
Let's define
  x  xL
  y  yL ,
  z  zL
  2V 
Vxx   2 
 x  L
  2V 
Vxy  

 xy  L
  2V 
Vyy   2 
 y  L
and
  2V 
Vzz   2 
 z  L
  2V 
Vyz  

 yz  L
  2V 
Vxz  

 xz  L
8
(11)
Combining these equations with Eq. (5), the linearised equations of motion relative to
the equilibrium point can be expressed as
  2  Vxx  Vxy  Vxz  0
  2  Vxy  Vyy  Vyz  0
(12)
  Vxz  Vyz  Vzz  0
It is natural to write the characteristic equation in the following form
 2  Vxx 2  Vxy
2  Vxy
 2  Vyy
Vxz
Vxz
Vyz  0
2
  Vzz
Vyz
(13)
Furthermore, it can be written as
 6  Vxx  Vyy  Vzz  4 2   4  VxxVyy  VyyVzz  VzzVxx  Vxy2  Vyz2  Vxz2  4 2Vzz   2
 VxxVyyVzz  2VxyVyzVxz  VxxVyz2  VyyVxz2  VzzVxy2   0
,
(14)
where  denotes the eigenvalues of Eq. (12). Eq. (14) is a sextic equation for  .
The stability of the equilibrium point is determined by six roots of Eq. (13). Let
i  i  1, 2, ,6  be the roots of Eq. (13). The equilibrium point is asymptotically
stable if and only if Re i  0 for i  1, 2,
,6 , i.e., the equilibrium point is a sink
(minimum) of the nonlinear dynamics system (5). Furthermore, the equilibrium point
is unstable if and only if there is a i equals to one of 1, 2,
,6 , such that Re i  0 ,
i.e., the equilibrium point is a source or a saddle.
3. Stability of the Equilibrium Points in the Potential Field of a Rotating
Asteroid
In this section, the stability of the equilibrium points in the potential field of a
9
rotating asteroid is studied. First, the roots of the characteristic equation at the
equilibrium point are investigated; then, a theorem about the eigenvalues is given. In
addition, a sufficient condition for the stability of equilibrium points is presented,
which only depends on the Hessian matrix of the effective potential. Finally, a
necessary and sufficient condition for the stability of equilibrium points is also
presented.
Let's denote
P      6  Vxx  Vyy  Vzz  4 2   4  VxxVyy  VyyVzz  VzzVxx  Vxy2  Vyz2  Vxz2  4 2Vzz   2
 VxxVyyVzz  2VxyVyzVxz  VxxVyz2  VyyVxz2  VzzVxy2 
, and it follows that P     P    . Hence:
Proposition 1. If  is an eigenvalue of the equilibrium point in the potential field of
a uniformly rotating asteroid, then  ,  , and  are also eigenvalues of the
equilibrium point. Namely, all eigenvalues are likely to have the form
   R,  0  , i    R,  0  , and   i  ,  R; ,  0  . □
 Vxx Vxy Vxz 


Theorem 1. If the matrix  Vxy Vyy Vyz  is positive definite, the equilibrium point
V

 xz Vyz Vzz 
in the potential field of a rotating asteroid is stable.
Proof:
Eq. (12) can be expressed as
MX  GX  KX  0 ,
where
10
(15)
X     
T
 Vxx Vxy Vxz 
1 0 0
 0 2 0 



, M   0 1 0  , G   2
0
0  , K   Vxy Vyy Vyz 
 0
0 0 1
V

0
0 



 xz Vyz Vzz 
The matrices M , G , and K satisfy MT  M  0 , G T  G , and K T  K ,
respectively. The system that can be expressed by the equation MX  GX  KX  0
 Vxx Vxy Vxz 


is stable if the matrix K   Vxy Vyy Vyz  is positive definite. □
V

 xz Vyz Vzz 
Theorem 1 is a sufficient condition for the stability of the equilibrium points in
the potential field of a rotating asteroid. The following theorem 2 will provide a
necessary and sufficient condition for the stability of the equilibrium points in the
potential field of a rotating asteroid.
Theorem 2. The equilibrium point in the potential field of a rotating asteroid is stable
if and only if
Vxx  Vyy  Vzz  4 2  0

2
2
2
2
VxxVyy  VyyVzz  VzzVxx  4 Vzz  Vxy  Vyz  Vxz
,

2
2
2
VxxVyyVzz  2VxyVyzVxz  VxxVyz  VyyVxz  VzzVxy
 2
3
3
2
 A  18 ABC  4 A C  4 B  27C
(16)
where
 A  Vxx  Vyy  Vzz  4 2


2
2
2
2
 B  VxxVyy  VyyVzz  VzzVxx  Vxy  Vyz  Vxz  4 Vzz

2
2
2

C  VxxVyyVzz  2VxyVyzVxz  VxxVyz  VyyVxz  VzzVxy
Proof:
Consider the linearised equation of motion relative to the equilibrium point
MX  GX  KX  0
Because the matrices M , G , K satisfy MT  M  0 , G T  G , K T  K , The
11
linearised equation is a gyroscope conservative system.
The characteristic equation of the gyroscope conservative system is
P      6  Vxx  Vyy  Vzz  4 2   4  VxxVyy  VyyVzz  VzzVxx  Vxy2  Vyz2  Vxz2  4 2Vzz   2
 VxxVyyVzz  2VxyVyzVxz  VxxVyz2  VyyVxz2  VzzVxy2   0
Using the stability condition of the gyroscope conservative system (Hughes 1986), the
conclusion can easily be obtained. □
From Theorem 2, it can be noted that the necessary and sufficient condition for
the stability of the equilibrium points in the potential field of a rotating asteroid
depends on the effective potential and the rotational angular velocity of the asteroid.
4 . Orbits, Differentiable Manifold and Geodesic
Previously, we denoted
 x, y, z 
as the position of the spacecraft in the asteroid
body-fixed coordinate system. Assuming that A3 is the topological space generated
 x, y, z  ,
by
the open sets of A3 are naturally defined. It is clear that A3 is a
smooth manifold. In this section, a metric d  2 is provided so that the geodesic of
A , d 
3
2
is the orbit of the particle relative to the asteroid.
Theorem 3. Denote d  2  2m  h  U  dr  dr   ω  r    ω  r  ; then, the orbit of
the particle relative to the asteroid on the manifold H  h is the geodesic of
A , d 
3
2
with the metric d  2 . In addition, the geodesic of
A , d 
3
2
with the
metric d  2 is the orbit of the particle relative to the asteroid on the manifold
H h.
Proof:
The equation of motion is Eq. (1), and the Jacobian constant is expressed by Eq.
12
(2).
Let
J    dr  dr   ω  r    ω  r  dt
t1
t2
(17)
Then, on the manifold H  h , the variation is zero.
J 0
On the manifold H  h , the following equation holds
dr dr
  ω  r   ω  r   2  h  U 
dt dt
(18)
Substituting this result into Eq. (17) yields the following
r2
J   d ,
r1
from which it can be shown that the variation  J along the orbit is zero. This result
implies that the orbit on the manifold H  h is the geodesic of
metric d  2 , and the geodesic of
A , d 
3
2
A , d 
3
2
with the
with the metric d  2 is the orbit on the
manifold H  h . □
Denote M   A3 , d  2  , where M is a smooth manifold, and the metric d  2
is not positive definite. For the equilibrium point L  M , denote its tangent space as
TL M . Then, dim M  dim TL M  3 . Let
TM 
Tp M 
pM
then, TM
 p, q  p  M , q T M  ;
p
is its tangent bundle. It follows that dim TM  6 . Let  be a
sufficiently small open neighborhood of the equilibrium point on the smooth manifold
M . Then, the tangent bundle of  is T  
Tp  
p
dim T  6 . Let
 S,  
 p, q  p , q T  , and
p
be a 6-dimensional symplectic manifold near the
13
equilibrium point such that S and T are topological homeomorphism but not
diffeomorphism, where  is a non-degenerate skew-symmetric bilinear quadratic
form.
5. Manifold, Periodic Orbits and Quasi-periodic Orbits near Equilibrium Points
To determine the motion, the manifold, the periodic orbits and the quasi-periodic
orbits near the equilibrium points, the stability and the eigenvalues of the equilibrium
points must be known.
5.1 Eigenvalues and Submanifold
The equilibrium point in the potential field of a rotating asteroid has 6
eigenvalues,
which
are
in
the
form
of
 j   R,  0; j  1, 2,3

,
i j    R,  0; j  1, 2  , and   i  ,  R; ,  0  . In fact, the forms of the
eigenvalues determine the structure of the submanifold and the subspace. There is a
bijection between the form of the eigenvalues and the submanifold or the subspace.
Let us denote the Jacobian constant at the equilibrium point by H  L  , and the
eigenvector of the eigenvalue  j as u j .
Let us define the asymptotically stable, the asymptotically unstable and the centre
manifold of the orbit on the manifold H  h near the equilibrium point, where
h  H  L    2 , and  2 is sufficiently small that there is no other equilibrium point
L in the sufficiently small open neighborhood on the manifold  S,   with the
 
Jacobian constant H L
 
that satisfies H  L   H L  h .
14
The asymptotically stable manifold W s  S  , the asymptotically unstable
manifold W u  S  , and the centre manifold W c  S  are tangent to the asymptotically
stable subspace E s  L  , the asymptotically unstable subspace E u  L  , and the centre
subspace E c  L  at the equilibrium point, respectively, where






E s  L   span u j Re  j  0 ,
E c  L   span u j Re  j  0 ,
E u  L   span u j Re  j  0 .
Let's define W r  S  as the resonant manifold, which is tangent to the resonant


subspace E r  L   span u j k , s.t.Re  j  Re k  0, Im  j  Im k , j  k .
Thus, it can be seen that
 S, 
T   W s  S   W c  S   W u  S  , where
denotes a topological homeomorphism,  denotes a diffeomorphism, and 
denotes a direct sum. Then, E r  L   E c  L  and W r  S   W c  S  .
By defining TLS as the tangent space of the manifold
homeomorphism
of
the
tangent
space
can
be
 S,  
, the
written
as
TLS  E s  L   E c  L   E u  L  . Considering the dimension of the manifolds, the
following equations hold
dimW s  S   dimW c  S   dimW u  S   dim  S,    dim T   6
dim E s  L   dim E c  L   dim E u  L   dim TLS  6
dim E r  L   dimW r  S   dim E c  L   dimW c  S 
Based on the conclusions above, the following theorem can be obtained.
Theorem 4. There are eight cases for the equilibrium points in the potential field of a
rotating asteroid:
15
Case
1:
The
eigenvalues
i j   j  R, j  0; j  1, 2,3
 S, 

are
different
and
in
the
form
of
; then, the structure of the submanifold is
T   W c  S  , and dimW r  S   0 .
 j  j  R, j  0, j  1  and
Case 2: The forms of the eigenvalues are
i j   j  R, j  0; j  1, 2  , and the imaginary eigenvalues are different; then, the
structure
of
the
submanifold
is
 S, 
T   W s S W c S W u S ,
dimW c  S   4 , dimW s  S   dimW u  S   1 , and dimW r  S   0 .
Case 3: The forms of the eigenvalues are  j  j  R, j  0; j  1, 2
i j   j  R, j  0, j  1  ;
 S, 
then,
the
T   W s S W c S W u S
structure
,
of
the
dimW r  S   0

submanifold
,
and
is
and
dimW s  S   dimW c  S   dimW u  S   2 .
Case 4: The forms of the eigenvalues are  j  j  R, j  0, j  1  and
  i  ,  R; ,  0
 S, 

;
then,
the
structure
of
the
submanifold
is
T   W s  S   W u  S  , and dimW s  S   dimW u  S   3 .
Case 5: The forms of the eigenvalues are i j   j  R, j  0, j  1  and
  i  ,  R; ,  0
 S, 

;
then,
the
structure
T   W s S W c S W u S
,
of
the
dimW r  S   0
submanifold
,
is
and
dimW s  S   dimW c  S   dimW u  S   2 .
Case 6: The forms of the eigenvalues are i j   j  R,1  2  3  0; j  1, 2,3  ;
then, the structure of the submanifold is
 S, 
T  W c  S  W r  S  , and
dimW r  S   dimW c  S   6 .
Case
7:
The
forms
of
16
the
eigenvalues
are
i j   j  R, j  0, 1  2  3 ; j  1, 2,3  ; then, the structure of the submanifold is
 S, 
T   W c  S  , and dimW r  S   4 .
Case
8:
The
forms
of
the
eigenvalues
are
 j  j  R, j  0, j  1  , i j   j  R,1  2  0; j  1, 2  ; then, the structure of
the
submanifold
is
 S, 
T   W s S W c S W u S
,
dimW r  S   dimW c  S   4 , and dimW s  S   dimW u  S   1 . □
Theorem 4 describes the structure of the submanifold as well as the stable and
unstable behaviours of the particle near the equilibrium points. For Cases 6-8, because
the resonant manifold and the resonant subspace exist, the equilibrium point is
resonant. Considering that the structures of the submanifold and the subspace are
fixed by the characteristic of the equilibrium points, it can be concluded that the
equilibrium points with resonant manifolds are resonant equilibrium points. Only
Case 1 leads to linearly stable equilibrium points, and Cases 2-5 lead to unstable
equilibrium points. Thus, one can obtain
Corollary 1. The equilibrium point is linearly stable if and only if it belongs to Case 1.
The equilibrium point is unstable and non-resonant if and only if it belongs to one of
the Cases 2-5. The equilibrium point is resonant if and only if it belongs to one of the
Cases 6-8.
The classes of orbit near the equilibrium points include: periodic orbit, Lissajous
orbit, quasi-periodic orbit, almost periodic orbit, etc. Let T k be a k -dimensional
torus. Then, the periodic orbit is on a 1-dimensional torus T 1 , the Lissajous orbit is
on a 2-dimensional torus T 2 , and the quasi-periodic orbit is on a k-dimensional torus
17
T k  k  1 .
5.2 Linearly Stable Equilibrium Points
Theorem 4 and Corollary 1 show that the linearly stable equilibrium points only
correspond to Case 1. In this section, more properties of Case 1 are discussed.
In Case 1, there are three pairs of imaginary eigenvalues of the equilibrium point,
which is linearly stable. The motion of the spacecraft relative to the equilibrium point
is a quasi-periodic orbit, which is expressed as
  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t (19)

  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t
There are three families of periodic orbits, which have periods
T1 
2
1
, T2 
2
2
, T3 
2
(20)
3
With the condition
C 2  C 2  C 2  S 2  S 2  S 2  C 3  C 3  C 3  S 3  S 3  S 3  0 , the first
family of periodic orbits has the form
  C 1 cos 1t  S 1 sin 1t

  C1 cos 1t  S1 sin 1t

  C 1 cos 1t  S 1 sin 1t
The period of the first family of periodic orbits is T1 
(21)
2
1
.
Denote t0 as the initial time. Then, the initial state can be expressed as
  t0    0,   t 0   0

  t0    0,  t 0   0,

  t0    0,   t 0   0
18
(22)
and the coefficients yield

0
C 1   0 cos 1t0  sin 1t
1



 S 1   0 sin 1t0  0 cos 1t
1



C 1  0 cos 1t0  0 sin 1t
1


 S   sin  t  0 cos  t
0
1 0
1
 1
1

C   cos  t   0 sin  t
0
1 0
1
 1
1


0
 S 1   0 sin 1t0  cos 1t
1

(23)
Using Eq. (23), we can calculate the coefficients of the first families of periodic
orbits if we know the initial state. The other two families of periodic orbits satisfy the
condition
C 1  C1  C 1  S 1  S1  S 1  C 3  C 3  C 3  S 3  S 3  S 3  0 , or
C 1  C1  C 1  S 1  S1  S 1  C 2  C 2  C 2  S 2  S 2  S 2  0
In addition, they have similar forms of the position equation and the coefficient
equation.
Theorem 5. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) It is linearly stable.
b) The
roots
of
the
characteristic
of i j   j  R, j  0; j  1, 2,3

equation
, and if
P  
are
in
the
form
j  k  j  1, 2,3; k  1, 2,3 , then
 j  k .
c) The motion of the spacecraft relative to the equilibrium point is a quasi-periodic
19
orbit, which is expressed as
  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t
d) There are four families of quasi-periodic orbits in the tangent space of the
equilibrium point, which can be expressed as
  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t

  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t

  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t
  C 1 cos 1t  S 1 sin 1t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  C 3 cos 3t  S 3 sin 3t

  C 1 cos 1t  S 1 sin 1t  C 3 cos 3t  S 3 sin 3t
  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t
  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t

  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t  C 3 cos 3t  S 3 sin 3t
e) There are four families of quasi-periodic orbits near the equilibrium point, and
they are on the k-dimensional tori T k  k  2,3 .
f) There are three families of periodic orbits in the tangent space of the equilibrium
point, which have the periods T1 
2
1
, T2 
2
2
, T3 
2
3
.
g) There are three families of periodic orbits near the equilibrium point.
h) There is no asymptotically stable manifold near the equilibrium point, and the
roots of the characteristic equation P    satisfy the following condition: If
j  k  j  1, 2,
,6; k  1, 2,
,6  , then  j  k .
i) There is no unstable manifold near the equilibrium point, and the roots of the
20
characteristic
equation
j  k  j  1, 2,
,6; k  1, 2,
P  
satisfy
the
following
condition:
If
,6  , then  j  k .
j) The dimensions of the centre manifold W c  S  and the resonant manifold
W r  S  satisfy dimW c  S   6 and dimW r  S   0 , respectively.
k) The dimensions of the centre subspace E c  L  and the resonant subspace
E r  L  satisfy dim E c  L   6 and dim E r  L   0 , respectively.
l) The structure of the submanifold is
 S, 
T   W c  S  , W r  S    , where
 is an empty set.
m) The structure of the subspace is TLS  E c  L  , and E r  L    .
Proof:
a)  b): Using Theorem 4, it is obvious.
b)  c): Using the solution theory of ordinary differential equations, one can obtain
c).
c)  d): It is obvious.
d)  e): Among the four families of quasi-periodic orbits that are expressed in d), the
first three families of quasi-periodic orbits near the equilibrium point are on 2dimensional tori T 2 , and the last family of quasi-periodic orbits near the equilibrium
3
point is on a 3- dimensional torus T .
e)  f): Assuming that there are four families of quasi-periodic orbits near the
equilibrium point, which are on the k-dimensional tori T k  k  2,3 , we can eliminate
Cases 2-8. Thus, the equilibrium point is linearly stable, and a) is established.
Considering a) and c), we derive that there are three families of periodic orbits in the
21
tangent space of the equilibrium point that have the periods T1 
T3 
2
3
2
1
, T2 
2
2
and
.
f)  g): It is obvious.
g)  a): Because there are three families of periodic orbits, we can eliminate Cases
2-8 and obtain a).
a)  i): Because the equilibrium point is linearly stable, there is no unstable manifold
or resonant manifold near the equilibrium point. We obtain i).
h)  i)  j)  k)  l)  m): It is obvious.
m)  a): The structure of the subspace is TLS  E c  L  , and E r  L    . Thus, the
dimensions of the unstable manifold and the resonant manifold near the equilibrium
point are zero. Then, the eigenvalues have the form i j   j  R, j  0; j  1, 2,3  ,
and if j  k  j  1, 2,3; k  1, 2,3 , then  j   k . This result leads to a). □
Strictly speaking, Eq. (19) is the projection of the motion to the tangent space of the
equilibrium point; Eq. (19) is the approximate expression of the motion. Because
Theorem 6 is about the necessary and sufficient conditions of the linear stability of the
equilibrium points in the potential field of a rotating asteroid, there is a corollary for
the linear instability of the equilibrium points.
Corollary 2. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) It is linearly unstable.
b) The roots of the characteristic equation P    do not have the form
22
i j   j  R, j  0;  j  k ; j, k  1, 2,3, j  k  .
c) There are fewer than four families of quasi-periodic orbits on the k-dimensional
tori T k  k  2,3 near the equilibrium point.
d) There are fewer than three families of periodic orbits in the tangent space of the
equilibrium point.
e) There is an asymptotically stable manifold near the equilibrium point or the roots
of the characteristic equation P    that satisfies the following condition: If
j  k  j  1, 2,
,6; k  1, 2,
,6  , then  j  k .
f) There is an unstable manifold near the equilibrium point or the roots of the
characteristic equation
j  k  j  1, 2,
P  
,6; k  1, 2,
that satisfies the following condition: If
,6  , then  j  k .
dim W c  S   6
g) dimW  S   6 or 
.
r
dim W  S   0
c
dim E c  L   6
h) dim E  L   6 or 
. □
r
dim E  L   0
c
5.3 Unstable Equilibrium Points
In this section, the unstable equilibrium points with no resonant manifold are
discussed. The unstable equilibrium points can be classified into four cases, which
have been presented in Theorem 4. From Theorem 4, it is known that an equilibrium
point is unstable if and only if dimW u  S   1,dimW r  S   0 , with
Case 2 corresponding to dimW u  S   1 and dimW r  S   0 ;
Case 3 and Case 5 corresponding to dimW u  S   2 and dimW r  S   0 ;
23
Case 4 corresponding to dimW u  S   3 and dimW r  S   0 .
5.3.1 Case 2
There are two pairs of imaginary eigenvalues and one pair of real eigenvalues for
the unstable equilibrium point. The motion of the spacecraft near this equilibrium
point relative to this equilibrium point is expressed by
  A 1e1t  B 1e 1t  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t


t
 t
  A1e 1  B1e 1  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t

1t
1t

  A 1e  B 1e  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t
(24)
The almost periodic orbit near the equilibrium point can be expressed by
  B 1e1t  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t


 t
  B1e 1  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t

1t

  B 1e  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t
(25)
The central manifold is a 4-dimensional smooth manifold, which is given by
A 1  B 1  A1  B1  A 1  B 1  0 . The motion of the spacecraft relative to the
equilibrium point on the central manifold is a Lissajous orbit.
There are two families of periodic orbits on the central manifold. The first family of

 A 1  B 1  A1  B1  A 1  B 1  0
periodic orbits is given by 
, and the period
C

S

C

S

C

S

0


2

2

2

2

2

2

is T1 
2
1
.
The
second
family
of
periodic
orbits

2
 A 1  B 1  A1  B1  A 1  B 1  0
, and the period is T2 
.

2

C 1  S 1  C1  S1  C 1  S 1  0
The asymptotically stable manifold is generated by
24
is
given
by
  B 1e 1t

 t
  B1e 1

1t
  B 1e
(26)
It is a 1-dimensional smooth manifold.
The unstable manifold is generated by
  A 1e1t

t
  A1e 1

1t
  A 1e
(27)
It is a 1-dimensional smooth manifold. The general result of Case 2 is stated as
follows.
Theorem 6.
For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The roots of the characteristic equation
P  
are in the form of
 j  j  R, j  0, j  1  and i j   j  R, j  0; j  1, 2  , where 1   2 .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
  A 1e1t  B 1e 1t  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t


t
 t
  A1e 1  B1e 1  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t

1t
1t

  A 1e  B 1e  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t
c) The characteristic roots are different, and there are two families of quasi-periodic
orbits in the tangent space of the equilibrium point, which can be expressed as
  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t

  C1 cos 1t  S1 sin 1t  C 2 cos  2t  S 2 sin  2t

  C 1 cos 1t  S 1 sin 1t  C 2 cos  2t  S 2 sin  2t
d) The characteristic roots are different, and there are two families of quasi-periodic
25
orbits near the equilibrium point on the 2-dimensional tori T 2 .
e) There are two families of periodic orbits in the tangent space of the equilibrium
point, which have the periods T1 
2
1
, T2 
2
2
.
f) There are two families of periodic orbits near the equilibrium point.
g) The structure of the submanifold is
 S, 
T   W s S W c S W u S ;
dimW c  S   4 , dimW s  S   dimW u  S   1 , and dimW r  S   0 .
h) The structure of the subspace is TLS  E s  L   E c  L   E u  L  ; dim E c  L   4 ,
dim E s  L   dim E u  L   1 , and dimW r  S   0 .
Proof:
The proof for Theorem 6 is similar to that for Theorem 5. □
5.3.2 Case 3
The
eigenvalues
of
 j  j  R, j  0; j  1, 2
the

equilibrium
point
are
in
the
form
of
and i j   j  R, j  0, j  1  . The equilibrium
point is unstable. The motion of the spacecraft near the equilibrium point relative to
the equilibrium point is expressed as
  A 1e1t  B 1e1t  A 2e 2t  B 2e  2t  C 1 cos 1t  S 1 sin 1t


t
 t
 t
 t
  A1e 1  B1e 1  A 2e 2  B 2e 2  C1 cos 1t  S1 sin 1t

1t
1t
 2t
 2 t

  A 1e  B 1e  A 2e  B 2e  C 1 cos 1t  S 1 sin 1t
(28)
The almost periodic orbit near the equilibrium point can be expressed as
  B 1e1t  B 2e  2t  C 1 cos 1t  S 1 sin 1t


 t
 t
  B1e 1  B 2e 2  C1 cos 1t  S1 sin 1t

1t
 2 t

  B 1e  B 2e  C 1 cos 1t  S 1 sin 1t
26
(29)

 A 1  B 1  A1  B1  A 1  B 1  0
The central manifold is given by 
, and it
A

B

A

B

A

B

0


2

2

2

2

2

2

is a 2-dimensional smooth manifold.
There
is
one
family
of
periodic
orbits,
which
is
given
by

2
 A 1  B 1  A1  B1  A 1  B 1  0
, and the period is T1 
.

A

B

A

B

A

B

0



2

2

2

2

2

2
1

The asymptotically stable manifold is generated by
  B 1e1t  B 2 e  2t

 t
 t
  B1e 1  B 2 e 2

1t
 2 t
  B 1e  B 2e
(30)
It is a 2-dimensional smooth manifold.
The unstable manifold is generated by
  A 1e1t  A 2 e 2t


t
 t
  A1e 1  A 2 e 2

1t
 2t

  A 1e  A 2e
(31)
It is a 2-dimensional smooth manifold. Then, the general result for Case 3 is stated as
follows.
Theorem 7. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The
roots
of
the
characteristic
 j  j  R, j  0; j  1, 2

equation
P  
have
the
form
and i j   j  R, j  0, j  1  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
27
  A 1e1t  B 1e1t  A 2e 2t  B 2e  2t  C 1 cos 1t  S 1 sin 1t


t
 t
 t
 t
  A1e 1  B1e 1  A 2e 2  B 2e 2  C1 cos 1t  S1 sin 1t

1t
1t
 2t
 2 t

  A 1e  B 1e  A 2e  B 2e  C 1 cos 1t  S 1 sin 1t
c) There is one family of periodic orbits in the tangent space of the equilibrium point,
which has the period T1 
2
1
, and the dimension of the unstable manifold
satisfies dimW u  S   2 ; there is at least one characteristic root in the real axis.
d) There is one family of periodic orbits near the equilibrium point, and the
dimension of the unstable manifold satisfies dimW u  S   2 ; there is at least one
characteristic root in the real axis.
e) The asymptotically stable manifold is generated by
  B 1e1t  B 2 e  2t

 t
 t
  B1e 1  B 2 e 2 .

1t
 2 t
  B 1e  B 2e
f) The unstable manifold is generated by
  A 1e1t  A 2 e 2t


t
 t
  A1e 1  A 2 e 2 .

1t
 2t

  A 1e  A 2e
g) The structure of the submanifold is
 S, 
T   W s  S   W c  S   W u  S  , and
dimW s  S   dimW c  S   dimW u  S   2 ; there is at least one characteristic root
in the real axis.
h) The structure of the subspace is
TLS  E s  L   E c  L   E u  L  , and
dim E s  L   dim E c  L   dim E u  L   2 ; there is at least one characteristic root
in the real axis. □
28
5.3.3 Case 4
The
forms
of
the
eigenvalues
are
 j  j  R, j  0, j  1 
and
  i  ,  R; ,  0  . The equilibrium point is unstable. The motion of the
spacecraft near the equilibrium point relative to the equilibrium point is expressed as
  A 1e1t  B 1e1t  E e1t cos 1t  F e1t sin  1t  G e 1t cos 1t  H  e 1t sin  1t

t
 t
t
t
 t
 t
  A1e 1  B1e 1  E e 1 cos 1t  F e 1 sin  1t  G e 1 cos 1t  H e 1 sin  1t (32)

1t
1t
1t
1t
1t
1t
  A 1e  B 1e  E e cos 1t  F e sin  1t  G e cos 1t  H  e sin  1t
The asymptotically stable manifold is generated by
  B 1e1t  G e 1t cos 1t  H  e 1t sin  1t

 t
 t
 t
  B1e 1  G e 1 cos 1t  H e 1 sin  1t

1t
1t
1t
  B 1e  G e cos 1t  H  e sin  1t
(33)
It is a 3-dimensional smooth manifold. The unstable manifold is generated by
  A 1e1t  E e1t cos 1t  F e1t sin  1t


t
t
t
  A1e 1  E e 1 cos  1t  F e 1 sin  1t

1t
1t
1t

  A 1e  E e cos 1t  F e sin  1t
(34)
It is a 3-dimensional smooth manifold. Then, the general result of Case 4 is stated as
follows.
Theorem 8. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The roots of the characteristic equation
P  
are in the form of
 j  j  R, j  0, j  1  and   i  ,  R; ,  0  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
29
  A 1e1t  B 1e1t  E e1t cos 1t  F e1t sin  1t  G e 1t cos 1t  H  e 1t sin  1t


t
 t
t
t
 t
 t
  A1e 1  B1e 1  E e 1 cos 1t  F e 1 sin  1t  G e 1 cos 1t  H e 1 sin  1t

1t
1t
1t
1t
1t
1t

  A 1e  B 1e  E e cos 1t  F e sin  1t  G e cos 1t  H  e sin  1t
c) There is no periodic orbit in the tangent space of the equilibrium point.
d) There is no periodic orbit near the equilibrium point.
e) The asymptotically stable manifold is generated by
  B 1e1t  G e 1t cos 1t  H  e 1t sin  1t


 t
 t
 t
  B1e 1  G e 1 cos 1t  H e 1 sin  1t .

1t
1t
1t

  B 1e  G e cos 1t  H  e sin  1t
f) The unstable manifold is generated by
  A 1e1t  E e1t cos 1t  F e1t sin  1t


t
t
t
  A1e 1  E e 1 cos  1t  F e 1 sin  1t .

1t
1t
1t

  A 1e  E e cos 1t  F e sin  1t
g) The structure of the submanifold is
 S, 
T   W s S W u S .
h) The dimension of the unstable manifold satisfies dimW u  S   3 .
i) The dimension of the asymptotically stable manifold satisfies dimW s  S   3
j) The structure of the subspace is TLS  E s  L   E u  L  .
k) The dimension of the unstable subspace satisfies dim E u  L   3 .
l) The dimension of the asymptotically stable subspace satisfies dim E s  L   3 . □
5.3.4 Case 5
The
forms
of
the
eigenvalues
are
i j   j  R, j  0, j  1 
and
  i  ,  R; ,  0  . The equilibrium point is unstable. The motion of the
spacecraft near the equilibrium point relative to the equilibrium point is expressed as
30
  C 1 cos 1t  S 1 sin 1t  E e1t cos 1t  F e1t sin  1t  G e 1t cos 1t  H  e 1t sin  1t

t
t
 t
 t
  C1 cos 1t  S1 sin 1t  E e 1 cos 1t  F e 1 sin  1t  G e 1 cos 1t  H e 1 sin  1t (35)

1t
1t
1t
1t
  C 1 cos 1t  S 1 sin 1t  E e cos 1t  F e sin  1t  G e cos 1t  H e sin  1t
The almost periodic orbit near the equilibrium point can be expressed as
  C 1 cos 1t  S 1 sin 1t  G e 1t cos 1t  H  e 1t sin  1t

 t
 t
  C1 cos 1t  S1 sin 1t  G e 1 cos 1t  H e 1 sin  1t

1t
1t
  C 1 cos 1t  S 1 sin 1t  G e cos 1t  H  e sin  1t
(36)
The central manifold is given by
 E  F  G  H   0

 E  F  G  H  0

 E  F  G  H   0
(37)
It is a 2-dimensional smooth manifold.
The asymptotically stable manifold is generated by
  G e1t cos 1t  H  e 1t sin  1t


 t
 t
  G e 1 cos 1t  H e 1 sin  1t

1t
1t

  G e cos 1t  H  e sin  1t
(38)
It is a 2-dimensional smooth manifold.
The unstable manifold is generated by
  E e1t cos 1t  F e1t sin  1t

t
t
  E e 1 cos 1t  F e 1 sin  1t

1t
1t
  E e cos 1t  F e sin  1t
(39)
It is a 2-dimensional smooth manifold. Then, the general result for Case 5 is stated as
follows.
Theorem 9. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
31
a) The roots of the characteristic equation
P    are in the form of
i j   j  R, j  0, j  1  and   i  ,  R; ,  0  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
  C 1 cos 1t  S 1 sin 1t  E e1t cos 1t  F e1t sin  1t  G e 1t cos 1t  H  e 1t sin  1t

t
t
 t
 t
  C1 cos 1t  S1 sin 1t  E e 1 cos 1t  F e 1 sin  1t  G e 1 cos 1t  H e 1 sin  1t

1t
1t
1t
1t
  C 1 cos 1t  S 1 sin 1t  E e cos 1t  F e sin  1t  G e cos 1t  H e sin  1t
c) There is only one family of periodic orbits in the tangent space of the equilibrium
point, which has the period T1 
2
1
, and the dimension of the unstable manifold
satisfies dimW u  S   2 . There is no characteristic root in the real axis.
d) There is only one family of periodic orbits near the equilibrium point, and the
dimension of the unstable manifold satisfies dimW u  S   2 . There is no
characteristic root in the real axis.
e) The asymptotically stable manifold is generated by
  G e1t cos 1t  H  e 1t sin  1t


 t
 t
  G e 1 cos 1t  H e 1 sin  1t .

1t
1t

  G e cos 1t  H  e sin  1t
f) The unstable manifold is generated by
  E e1t cos 1t  F e1t sin  1t

t
t
  E e 1 cos 1t  F e 1 sin  1t .

1t
1t
  E e cos 1t  F e sin  1t
g) The structure of the submanifold is
 S, 
T   W s S W c S W u S ,
and dimW s  S   dimW c  S   dimW u  S   2 . There is no characteristic root in
32
the real axis.
h) The
structure
of
the
subspace
is
TLS  E s  L   E c  L   E u  L  ,
and dim E s  L   dim E c  L   dim E u  L   2 . There is no characteristic root in
the real axis. □
5.4 Resonant Equilibrium Points
The resonant manifold and the resonant orbits exist if and only if one of Cases
6-8 are established. An equilibrium point is resonant if and only if dimW r  S   0 ,
with
Case 6 corresponding to
 S, 
T  W c S W r S ,
dimW r  S   dimW c  S   6 ;
Case 7 corresponding to
 S, 
T  W c  S  , dimW c  S   6 and
dimW r  S   4 ; as well as
Case 8 corresponding to
 S, 
T  W s  S   W c  S   W u  S  , dimW r  S   dimW c  S   4 , and
dimW s  S   dimW u  S   1 .
5.4.1 Case 6
The motion of the spacecraft relative to the equilibrium point is on a 1:1:1
resonant manifold, which can be expressed as
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  P 2t 2 cos 1t  Q 2t 2 sin 1t


2
2
  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t  P 2t cos 1t  Q 2t sin 1t (40)

2
2

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  P 2t cos 1t  Q 2t sin 1t
33
The resonant manifold is a 6-dimensional smooth manifold. There is only one family
 P 1  Q 1  P 2  Q 2  0

of periodic orbits, which is given by  P1  Q1  P 2  Q 2  0 , and the period is

 P 1  Q 1  P 2  Q 2  0
T1 
2
1
. Then, the general result of Case 6 is stated as follows.
Theorem 10. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The roots of the characteristic equation
P  
are in the form of
i j   j  R,1  2  3  0; j  1, 2,3  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  P 2t 2 cos 1t  Q 2t 2 sin 1t


2
2
  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t  P 2t cos 1t  Q 2t sin 1t

2
2

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  P 2t cos 1t  Q 2t sin 1t
c) The structure of the submanifold is
d) The
resonant
manifold
 S, 
is
T   W c S  W r S .
a
6-dimensional
manifold:
dimW r  S   dimW c  S   6 .
e) The structure of the subspace is TLS  E c  L   E r  L  .
f) The resonant subspace is a 6-dimensional space: dim E c  L   dim E r  L   6 . □
5.4.2 Case 7
The motion of the spacecraft near the equilibrium point relative to the equilibrium
point is expressed as
34
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t  C 3 cos 3t  S 3 sin 3t (41)

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  C 3 cos 3t  S 3 sin 3t
There is a 1:1 resonant manifold, which is generated by
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t

  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t
(42)
The resonant manifold is a 4-dimensional smooth manifold. There are two families of
periodic
orbits.
The
first
family
of
periodic
orbits
is
given
by
 P 1  Q 1  C 3  S 3  0

2
. The second family of periodic
 P1  Q1  C 3  S 3  0 , and the period is T1 
1

 P 1  Q 1  C 3  S 3  0
 P 1  Q 1  C 1  S 1  0

2
orbits is given by  P1  Q1  C1  S1  0 , and the period is T3 
. Then, the

3

 P 1  Q 1  C 1  S 1  0
general result of Case 7 is stated as follows.
Theorem 11. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The roots of the characteristic equation
P  
are in the form of
i j   j  R, j  0, 1  2  3 ; j  1, 2,3  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  C 3 cos 3t  S 3 sin 3t

  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t  C 3 cos 3t  S 3 sin 3t

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t  C 3 cos 3t  S 3 sin 3t
c) There are two families of periodic orbits in the tangent space of the equilibrium
point, and dimW r  S   4 .
35
d) The structure of the submanifold is
 S, 
T   W c  S  , and dimW r  S   4 .
e) dimW c  S   6 and dimW r  S   4 .
f) The structure of the subspace is TLS  E c  L  , and dim E r  L   4 .
g) dim E c  L   6 and dim E r  L   4 . □
5.4.3 Case 8
The motion of the spacecraft near the equilibrium point relative to the
equilibrium point is expressed as
  A 1e1t  B 1e 1t  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t


t
 t
  A1e 1  B1e 1  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t

1t
1t

  A 1e  B 1e  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t
(43)
There is a 1:1 resonant manifold, which is generated by
  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t

  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t

  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t
(44)
The resonant manifold is a 4-dimensional smooth manifold. There is only one family
 A 1  B 1  P 1  Q 1  0

of periodic orbits, which is given by  A1  B1  P1  Q1  0 , and the period is

 A 1  B 1  P 1  Q 1  0
T1 
2
1
. The asymptotically stable manifold is generated by
  B 1e 1t

 t
  B1e 1

1t
  B 1e
It is a 1-dimensional smooth manifold. The unstable manifold is generated by
36
(45)
  A 1e1t

t
  A1e 1

1t
  A 1e
(46)
It is a 1-dimensional smooth manifold. Then, the general result of Case 8 is stated as
follows.
Theorem 12. For an equilibrium point in the potential field of a rotating asteroid, the
following conditions are equivalent:
a) The roots of the characteristic equation
P  
are in the form of
 j  j  R, j  0, j  1  and i j   j  R,1  2  0; j  1, 2  .
b) The motion of the spacecraft near the equilibrium point relative to the equilibrium
point can be expressed as
  A 1e1t  B 1e 1t  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t


t
 t
  A1e 1  B1e 1  C1 cos 1t  S1 sin 1t  P1t cos 1t  Q1t sin 1t

1t
1t

  A 1e  B 1e  C 1 cos 1t  S 1 sin 1t  P 1t cos 1t  Q 1t sin 1t
c) There is one family of periodic orbits in the tangent space of the equilibrium
point, and dimW r  S   4 .
d) The structure of the submanifold is
 S, 
T   W s S W c S W u S ,
and dimW r  S   4 .
e) dimW c  S   dimW r  S   4 .
f) dimW c  S   dimW r  S   4 and dimW s  S   dimW u  S   1 .
g) The structure of the subspace is
TLS  E s  L   E c  L   E u  L  , and
dim E r  L   4 .
h) dim E c  L   dim E r  L   4 and dim E s  L   dim E u  L   1 . □
37
6. Applications to Asteroids
In this section, the theorems described in the previous sections are applied to
asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia, and 6489 Golevka. The
physical model of these four asteroids that we used here was calculated with radar
observations using the polyhedral model (Neese 2004).
6.1 Application to Asteroid 216 Kleopatra
The estimated bulk density of asteroid 216 Kleopatra is 3.6
g  cm3
(Descamps et al. 2011), and the rotation period of asteroid 216 Kleopatra is 5.385 h
(Ostro et al. 2000). Table 1 shows the positions of the equilibrium points in the
body-fixed frame, which were calculated by Newton method using Eq. (9). Table 2
shows the eigenvalues of the equilibrium points, which were calculated using Eq. (14).
E1 and E2 belong to Case 2, whereas E3 and E4 belong to Case 5. Yu & Baoyin
(2012a) calculated the positions of the equilibrium points as well as the eigenvalues of
the equilibrium points for asteroid 216 Kleopatra using the numerical method.
Table 1 Positions of the Equilibrium Points around Asteroid 216 Kleopatra
Equilibrium Points
E1
E2
E3
E4
x (km)
142. 852
-144.684
2.21701
-1.16396
y (km)
2.45436
5.18855
-102.102
100.738
z (km)
1.18008
-0.282998
0.279703
-0.541516
Table 2 Eigenvalues of the Equilibrium Points around Asteroid 216 Kleopatra
103
1
2
3
38
4
5
6
E1
E2
E3
0.376
0.422
0.327i
-0.376
-0.422
-0.327i
E4
0.323i
-0.323i
0.413i
0.414i
0.202+0.
304i
0.202+0.
306i
-0.413i
-0.414i
0.202-0.3
04i
0.202-0.3
06i
0.425i
0.466i
-0.202+0.
304i
-0.202+0.
306i
-0.425i
-0.466i
-0.202-0.
304i
-0.202-0.
306i
Figure 1.a A Quasi-Periodic Orbit near the Equilibrium Point E1
39
Figure 1.b
A Periodic Orbit near the Equilibrium Point E1
40
Figure 2.
A Quasi-Periodic Orbit near the Equilibrium Point E2
Figure 1.a shows a quasi-periodic orbit near the equilibrium point E1, where the
41

C 1  C 1  S1  C 2  10
coefficients have the values 
and other coefficients being
S

2


2

equal to zero. The flight time of the orbit is 12 days. Figure 1.b shows a periodic orbit
near the equilibrium point E1, where the coefficients have the values
C 1  C 1  S1  10 and other coefficients being equal to zero. E1 belongs to Case 2.
There is one family of quasi-periodic orbits near E1, which is on the 2-dimensional
tori T 2 . The orbit in figure 1 belongs to the quasi-periodic orbital family. For the
equilibrium point E2, figure 2 shows a quasi-periodic orbit near the equilibrium point
E2, the coefficients have the values

C 1  C 1  S1  C 2  10


 S 2  2
and other
coefficients being equal to zero. The flight time of the orbit is 12 days. E2 also
belongs to Case 2. There is one family of quasi-periodic orbits near E2, which is on
2
the 2- dimensional tori T . The orbit belongs a quasi-periodic orbital family that is
notably similar to that of E1.
42
Figure 3. An Orbit near the Equilibrium Point E3
43
Figure 4. An Orbit near the Equilibrium Point E4
Figure 3 shows an almost periodic orbit near the equilibrium point E3. The flight
time of the orbit is 12 days. There is only one family of periodic orbits near E3.
44
Figure 4 shows an almost periodic orbit near the equilibrium point E4.The flight time
of the orbit around the equilibrium point E4 is 12 days. There is only one family of
periodic orbits near E4.
6.2 Application to Asteroid 1620 Geographos
The estimated bulk density of asteroid 1620 Geographos is 2.0
g  cm3
(Hudson & Ostro 1999), and the rotation period of asteroid 1620 Geographos is
5.222h (Ryabova 2002). Table 3 shows the positions of the equilibrium points, which
were calculated using Eq. (9). Table 4 shows the eigenvalues of the equilibrium points.
E1 and E2 belong to Case 2, whereas E3 and E4 belong to Case 5.
Table 3 Positions of the Equilibrium Points around Asteroid 1620 Geographos
Equilibrium Points
E1
E2
E3
E4
x (km)
2.69925
-2.84097
-0.141618
-0.125678
y (km)
-0.041494
-0.057621
2.11961
-2.08723
z (km)
0.085296
0.142056
-0.021510
-0.025536
Table 4 Eigenvalues of the Equilibrium Points around Asteroid 1620 Geographos
103
1
2
3
4
5
6
E1
E2
E3
0.455
0.608
0.334i
-0.455
-0.608
-0.334i
E4
0.334i
-0.334i
0.427i
0.511i
0.152+0.
271i
0.174+0.
284i
-0.427i
-0.511i
0.152-0.2
71i
0.174-0.2
84i
0.487i
0.566i
-0.152+0.
271i
-0.174+0.
284i
-0.487i
-0.566i
-0.152-0.
271i
-0.174-0.
284i
45
Figure 5. A Quasi-Periodic Orbit near the Equilibrium Point E1
46
Figure 6. A Quasi-Periodic Orbit near the Equilibrium Point E2
47
Figure 5 shows a quasi-periodic orbit near the equilibrium point E1, where the

C 1  C 1  S1  C 2  0.01
coefficients have the values 
and other coefficients

 S 2  0.02
being equal to zero. The flight time of the orbit is 12 days. E1 belongs to Case 2.
There is one family of quasi-periodic orbits near E1, which is on the 2-dimensional
tori T 2 . The orbit in figure 5 belongs to the quasi-periodic orbital family. For the
equilibrium point E2, figure 6 shows a quasi-periodic orbit near the equilibrium point
E2, the coefficients have the values

C 1  C 1  S1  C 2  0.1
and other


 S 2  0.2
coefficients being equal to zero. The flight time of the orbit is 12 days. E2 also
belongs to Case 2. There is one family of quasi-periodic orbits near E2, which is on
2
the 2- dimensional tori T . The orbit belongs a quasi-periodic orbital family that is
notably similar to that of E1.
48
Figure 7. An Orbit near the Equilibrium Point E3
49
Figure 8. An Orbit near the Equilibrium Point E4
Figure 7 shows an almost periodic orbit near the equilibrium point E3. The flight
time of the orbit is 12 days. There is only one family of periodic orbits near E3.
Figure 8 shows an almost periodic orbit near the equilibrium point E4. The flight time
50
of the orbit around the equilibrium point E4 is 12 days. There is only one family of
periodic orbits near E4.
6.3 Application to Asteroid 4769 Castalia
The estimated bulk density of asteroid 4769 Castalia is 2.1 g  cm3 (Hudson &
Ostro 1994; Scheeres et al. 1996), and the rotation period of asteroid 4769 Castalia is
4.095h (Hudson et al. 1997). Table 5 shows the positions of the equilibrium points,
which were calculated using Eq. (9). Table 6 shows the eigenvalues of the equilibrium
points. E1 and E2 belong to Case 2, whereas E3 and E4 belong to Case 5.
Table 5 Positions of the Equilibrium Points around Asteroid 4769 Castalia
Equilibrium Points
E1
E2
E3
E4
x (km)
0.978767
-1.01550
-0.043617
-0.034338
y (km)
0.023603
0.116078
0.815747
-0.823895
z (km)
0.028032
0.0257862
0.002265
-0.006950
Table 6 Eigenvalues of the Equilibrium Points around Asteroid 4769 Castalia
103
1
2
3
4
5
6
E1
E2
E3
0.341
0.423
0.380i
-0.341
-0.423
-0.380i
E4
0.386i
-0.386i
0.416i
0.467i
0.195+0.
324i
0.197+0.
322i
-0.416i
-0.467i
0.195-0.3
24i
0.197-0.3
22i
0.470i
0.490i
-0.195+0.
324i
-0.197+0.
322i
-0.470i
-0.490i
-0.195-0.
324i
-0.197-0.
322i
51
Figure 9. A Quasi-Periodic Orbit near the Equilibrium Point E1
52
Figure 10. A Quasi-Periodic Orbit near the Equilibrium Point E2
Figure 9 shows a quasi-periodic orbit near the equilibrium point E1, where the
53

C 1  C 1  S1  C 2  0.04
coefficients have the values 
and other coefficients
S

0.08


2

being equal to zero. The flight time of the orbit is 4 days. E1 belongs to Case 2. There
is one family of quasi-periodic orbits near E1, which is on the 2-dimensional tori T 2 .
The orbit in figure 9 belongs to the quasi-periodic orbital family. For the equilibrium
point E2, figure 10 shows a quasi-periodic orbit near the equilibrium point E2, the

C 1  C 1  S1  C 2  0.04
coefficients have the values 
and other coefficients

 S 2  0.08
being equal to zero. The flight time of the orbit is 4 days. E2 also belongs to Case 2.
There is one family of quasi-periodic orbits near E2, which is on the 2- dimensional
2
tori T . The orbit belongs a quasi-periodic orbital family that is notably similar to
that of E1.
54
Figure 11. An Orbit near the Equilibrium Point E3
55
Figure 12. An Orbit near the Equilibrium Point E4
Figure 11 shows an almost periodic orbit near the equilibrium point E3. The
flight time of the orbit is 4 days. There is only one family of periodic orbits near E3.
Figure 12 shows an almost periodic orbit near the equilibrium point E4. The flight
time of the orbit around the equilibrium point E4 is 4 days. There is only one family
of periodic orbits near E4.
6.4 Application to Asteroid 6489 Golevka
The estimated bulk density of asteroid 6489 Golevka is 2.7 g  cm3 (Mottola
et al. 1997), and the rotation period of asteroid 6489 Golevka is 6.026h (Mottola et al.
1997). Table 7 shows the positions of the equilibrium points, which were calculated
using Eq. (9). Table 8 shows the eigenvalues of the equilibrium points. E1 and E2
56
belong to Case 2, whereas E3 and E4 belong to Case 1.
Table 7 Positions of the Equilibrium Points around Asteroid 6489 Golevka
Equilibrium Points
E1
E2
E3
E4
x (km)
0.564128
-0.571527
-0.021647
-0.026365
y (km)
-0.0234156
0.035808
0.537470
-0.546646
z (km)
-0.002882
-0.006081
-0.001060
-0.000182
Table 8 Eigenvalues of the Equilibrium Points around Asteroid 6489 Golevka
103
1
2
3
4
5
6
E1
E2
E3
E4
0.125
0.181
0.207i
0.185i
-0.125
-0.181
-0.207i
-0.185i
0.297i
0.304i
0.216i
0.213i
-0.297i
-0.304i
-0.216i
-0.213i
0.309i
0.329i
0.280i
0.297i
-0.309i
-0.329i
-0.280i
-0.297i
57
Figure 13. A Quasi-Periodic Orbit near the Equilibrium Point E1
58
Figure 14. A Quasi-Periodic Orbit near the Equilibrium Point E2
Figure 13 shows a quasi-periodic orbit near the equilibrium point E1, where the

C 1  C 1  S1  C 2  0.01
coefficients have the values 
and other coefficients

 S 2  0.02
being equal to zero. The flight time of the orbit is 4 days. E1 belongs to Case 2. There
is one family of quasi-periodic orbits near E1, which is on the 2-dimensional tori T 2 .
The orbit in figure 13 belongs to the quasi-periodic orbital family. For the equilibrium
point E2, figure 14 shows a quasi-periodic orbit near the equilibrium point E2, the

C 1  C 1  S1  C 2  0.01
coefficients have the values 
and other coefficients

 S 2  0.02
being equal to zero. The flight time of the orbit is 4 days. E2 also belongs to Case 2.
There is one family of quasi-periodic orbits near E2, which is on the 2- dimensional
2
tori T . The orbit belongs a quasi-periodic orbital family that is notably similar to
59
that of E1.
Figure 15. A Quasi-Periodic Orbit near the Equilibrium Point E3
60
Figure 16. A Quasi-Periodic Orbit near the Equilibrium Point E4
61
Figure 15 shows a quasi-periodic orbit near the equilibrium point E3, where the
C 1  0.03

coefficients have the values  S1  0.04 ,

C 1  0.01
C 2  0.005
,

 S 2  0.006
C 3  0.003
and other

 S 3  0.002
coefficients being equal to zero. The flight time of the orbit is 12 days. E3 belongs to
Case 1. There are four families of quasi-periodic orbits near E3, which are on the
3-dimensional tori T 3 . The orbit in figure 15 belongs to a quasi-periodic orbital
family. For the equilibrium point E4, figure 16 shows a quasi-periodic orbit near the
C 1  0.03

equilibrium point E4, the coefficients have the values  S1  0.04 ,

C 1  0.01
C 2  0.005
,

 S 2  0.006
C 3  0.003
and other coefficients being equal to zero. The flight time of the orbit is

 S 3  0.002
12 days. E4 also belongs to Case 1. There are four families of quasi-periodic orbits
near E4, which are on the 3- dimensional tori T 3 . The orbit belongs a quasi-periodic
orbital family that is notably similar to that of E3.
6.5 Discussion of Applications
The theorems described in the previous sections are applied to asteroids 216
Kleopatra, 1620 Geographos, 4769 Castalia, and 6489 Golevka.
For the asteroid 216 Kleopatra, 1620 Geographos, and 4769 Castalia, the
equilibrium points E1 and E2 belong to Case 2, whereas the equilibrium points E3
and E4 belong to Case 5. There are two families of periodic orbits and one family of
quasi-periodic orbits on the central manifold near each of the equilibrium points E1
and E2. There is only one family of periodic orbits on the central manifold near each
of the equilibrium points E3 and E4.
62
For the asteroid 6489 Golevka, E1 and E2 belong to Case 2, whereas E3 and E4
belong to Case 1. There are two families of periodic orbits and one family of
quasi-periodic orbits on the central manifold near each of the equilibrium points E1
and E2. The equilibrium points E3 and E4 around asteroid 6489 Golevka are linearly
stable. There are three families of periodic orbits and four families of quasi-periodic
orbits on the central manifold near each of the equilibrium points E3 and E4.
7. Conclusions
The motion of a particle in the potential field of a rotating asteroid is studied, and
the linearised equation of motion relative to the equilibrium points is presented. The
characteristic equation of equilibrium points is presented and discussed. In addition,
one sufficient condition and one necessary and sufficient condition for the stability of
the equilibrium points in the potential field of a rotating asteroid are provided.
Considering the orbit of the particle near the asteroid, we link the orbit and the
geodesic of a smooth manifold with a new metric. The metric does not have a positive
definite quadratic form. Then, we classify the equilibrium points into eight cases
using the eigenvalues. The structure of the submanifold near the equilibrium point is
related to the eigenvalues. A theorem is presented to describe the structure of the
submanifold as well as the stable and unstable behaviours of the particle near the
equilibrium points.
Near the linearly stable equilibrium point, there are four families of
quasi-periodic orbits, which are on the k-dimensional tori T k  k  2,3 . There are
neither asymptotically stable manifold nor asymptotically unstable manifold near the
63
linearly stable equilibrium point. The structures of the submanifold and the subspace
of the linearly stable equilibrium point are different from those of the linear unstable
equilibrium point.
Near the non-resonant unstable equilibrium points, the dimension of the unstable
manifold is greater than zero. The periodic orbit, the quasi-periodic orbit, etc., have
been studied near the non-resonant unstable equilibrium points. Near the resonant
equilibrium points, there is at least one family of periodic orbits. The dimension of the
resonant manifold is greater than four.
The theory is applied to asteroids 216 Kleopatra, 1620 Geographos, 4769
Castalia, and 6489 Golevka. For the asteroid 216 Kleopatra, 1620 Geographos, and
4769 Castalia, the equilibrium points are unstable, which are denoted as E1, E2, E3,
and E4, two of them belong to Case 2 while the other two of them belong to Case 5.
For the asteroid 6489 Golevka, two equilibrium points belong to Case 2; whereas the
other two equilibrium points belong to Case 1, i.e. they are linearly stable.
Acknowledgements
This research was supported by the National Basic Research Program of China (973
Program, 2012CB720000) and the National Natural Science Foundation of China (No.
11072122).
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